The White House was forced on to the back foot again on Monday over Donald Trump’s unsubstantiated claim that he was wiretapped by Barack Obama, basing its latest argument on the US president’s use of quotation marks in a tweet.

Pressured again on the issue at Monday’s White House briefing, press secretary Sean Spicer said: “He doesn’t really think that President Obama went up and tapped his phone personally. But I think there’s no question that the Obama administration, that there were actions about surveillance and other activities that occurred in the 2016 election. That is a widely reported activity that occurred back then.”

In a shift of position, Spicer then conceded: “The president used the word ‘wiretapped’ in quotes to mean broadly surveillance and other activities during that. It is interesting how many news outlets reported that this activity was taking place during the 2016 election cycle and now are wondering where the proof is. It is many of the same outlets in this room that talked about the activities that were going on back then.”

One of Trump’s tweets on 4 March did indeed use quotation marks when it said: “Terrible! Just found out that Obama had my ‘wires tapped’ in Trump Tower just before the victory. Nothing found. This is McCarthyism!”

But another tweet less than half an hour later did not use quotation marks: “How low has President Obama gone to tapp [sic] my phones during the very sacred election process. This is Nixon/Watergate. Bad (or sick) guy!”

Obama has strongly denied the claim, as has his director of national intelligence.

At Monday’s briefing, there were fractious exchanges between Spicer and the press over the veracity of Trump’s claims regarding unemployment numbers and health insurance statistics. Asked if, when the president says something, it can be trusted to be real, Spicer replied: “If he’s not joking, of course.” This prompted some laughter from the assembled press.

But tellingly, the questioning kept circling back to the wiretapping claim, suggesting that it could prove a vulnerable spot for the White House. Monday was the deadline for the Department of Justice to provide members of Congress with some evidence of Trump’s assertion.

Spicer said he had spoken to Trump directly about the tweets in question. “His tweets do speak for themselves ... He literally had it in quotes.”

But asked to be more precise about the nature of the alleged surveillance, the spokesman replied: “There’s a whole host of tactics that can be used to monitor somebody either through wiretap or other ways in which you can surveil somebody.”

He then wrapped up the briefing almost immediately before more questions could be asked.

Trump held his first cabinet meeting on Monday, with negotiations over the repeal and replacement of Obama’s Affordable Care Act still dominating the political agenda and a budget proposal expected to be unveiled on Thursday.

Spicer announced that Trump’s meeting with German chancellor Angela Merkel, scheduled for Tuesday, has been postponed until Friday due to the forecast of a snowstorm about to batter Washington and the north-east.

He acknowledged that planning is under way for a meeting with Chinese president Xi Jinping, with topics including the defusing of tensions with North Korea, but did not confirm reports that it will be held at Trump’s Mar-a-Lago estate in Florida early in April.

Spicer also had an expected announcement. Asked about billionaire Trump’s pledge to donate his presidential salary to charity, the press secretary said: “The president’s intention right now is to donate his salary at the end of the year and he has kindly asked that you all help determine where that goes. The way that we can avoid scrutiny is to let the press corps determine where it should go.”

The room erupted in laughter, but Spicer explained: “In all seriousness, I think his view is he made a pledge to the American people he wants to donate it to charity and he’d love your help to determine where it should go.”